1D Schrödinger operators with Coulomb-like potentials

We study the convergence of 1D Schrödinger operators Hε with the potentials which are regularizations of a class of pseudopotentials having, in particular, the form αδ′(x) + βδ(x) + γ/|x| or αδ′(x) + βδ(x) + γ/x. The limit behavior of Hε in the norm resolvent topology, as ε → 0, essentially depends on a way of regularization of the Coulomb potential and the existence of zero-energy resonances for δ′-like potential. All possible limits are described in terms of point interactions at the origin. As a consequence of the convergence results, different kinds of L∞(R)-approximations to the even and odd Coulomb potentials, both penetrable and impenetrable in the limit, are constructed.